This paper explores the parallel complexity of finite constraint satisfaction problems (FCSPs) by developing three algorithms for deriving minimal constraint networks in parallel. The first is a parallel algorithm for the EREW PRAM model, the second is a distributed algorithm for fine-grain interconnected networks, and the third is a distributed algorithm for coarse-grain interconnected networks. Our major results are: given an FCSP represented by an acyclic constraint network (or a join tree) of size n with treewidth bounded by a constant, then (1) the parallel algorithm takes O(log n) time using O(n) processors, (2) there is an equivalent network, of size poly(n) with treewidth also bounded by a constant, which can be solved by the fine-grain distributed algorithm in O(\log n) time using poly(n) processors and (3) the distributed algorithm for coarse-grain interconnected networks has linear speedup and linear scaleup. In addition, we have simulated the fine-grain distributed algorithm based on the logical time assumption, experimented with the coarse-grain distributed algorithm on a network of transputers, and evaluated the results against the theory.